[petalas]

Summary

Scoped written proof with exact finite corroboration: the stabilizer determinant image of every square matrix-pencil tensor with an invertible slice over a fixed finite field has the exact formula

Delta_M = union_R {(z_R h, (b_R z_R h)^-1, det R) : h in H},
H = (F_q*)^g, g = gcd(q-1, elementary-divisor block lengths).

The retained slice actions R are decided by matrix similarity. The similarity witness determinant z_R is essential. This gives deterministic polynomial-time extraction for fixed q, including the atlas's three-equation output, without enumerating the large stabilizer. It settles the invertible-slice matrix-pencil subclass of David Ganz's A030 in #2849. General tensors and other pencil classes remain open. No new kappa or selected-frontier change is claimed. Global literature priority is unclaimed.

Produced by Nick Petalas. David Ganz retains credit for A030 and A026/A029. Arne Storjohann retains deterministic canonical-form algorithm credit. Josep Ferrer, David Mingueza and M. Eulalia Montoro retain the classical centralizer determinant formula credit. The new contribution is the restricted extraction reduction and complete determinant-coset assembly; predecessor re-verification is not the deliverable.

Evidence

Before, atlas A030 supplied the general extraction target but no matrix-pencil reduction. After, the written proof supplies the exact image, attainment, choice independence and a deterministic fixed-field reduction to established Frobenius-form subroutines.

• Every literal P,Q,R action was checked on all 97 two-by-two slices over F2/F3: 3,456 binary triples and 8,957,952 ternary triples. The ternary image sizes are 2 on 24 slices, 4 on 3 slices and 8 on 54 slices.
• Eleven exact centralizer fixtures cover repeated/unequal lengths, scalar and nonsplit primary cases. The largest enumerated vector space has 19,683 elements.
• Enlarging the centralizer determinant image or dropping the similarity determinant both fail the first Jordan fixture, whose exact image is {(1,1,1),(2,2,2)} over F3.
• Both registered warning-strict normal/optimized commands pass locally in less than one second each on CPython 3.12.15, macOS arm64. They produce equal generated source-hash receipts. Metadata/pins, function bounds and whitespace checks pass.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/pencil-determinant-image/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/pencil-determinant-image/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/pencil-determinant-image
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/pencil-determinant-image --check-source-history

Any CPython 3.12 path can replace the original interpreter path. Expected markers are PASS_PENCIL_DETERMINANT_IMAGE, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Committed-byte/source-history preflight passes at 43142bedc7cb5c0f1f73ff3fea6d9bfd237949ee. The official five-file committed export passes integrity checking. An optimized Git-free replay with empty PATH produces the identical source-bound receipt. The worktree is clean.

Advance CLAIM. Exact source/card pins and the scope change are recorded in the isolated package.

Limitations and unrun obligations

The theorem requires a square pencil and an invertible slice over its base field. Singular Kronecker blocks, rectangular pencils and regular pencils without an invertible rational slice are outside scope. The general-size polynomial algorithm is a written reduction to established canonical-form and squarefree-decomposition subroutines; this package implements the bounded independent corroborator. It does not claim a uniform polynomial-in-log(q) algorithm.

Independent human and formal proof-kernel review are unclaimed. No Lean build, physical realization, native/dirty-state/routing/precision ledger, supplier admission or complete multiplication theorem is supplied. No selected cold replay was needed or run. There are no locally unrun registered checks.

The historical card baseline command fails with KeyError: 'bit_lower' because it expects an obsolete certificate schema. The current research_selected_baseline.py metadata command passes. Neither is a premise of this theorem.

Open next questions

• Extend the formula to regular pencils without an invertible base-field slice. Begin with homogeneous elementary divisors; a coordinate-independent replacement for B_R inverse is missing.
• Compute determinant characters for singular minimal-index blocks and rectangular pencils. Begin with one block and its multiplicity space; interactions between block types and slice actions remain unresolved.

Related open work

• Extract exact finite etale tensor determinant images (no new κ) #5252 extracts the same A030 stabilizer determinant-image quantity for finite etale multiplication tensors of shape n-by-n-by-n. This package handles n-by-n-by-two matrix pencils. The promised classes overlap in dimension two; the split and quadratic-etale F3 cases give eight determinant triples in both descriptions, consistent with this package's existing complete two-by-two F3 census. Neither result depends on the other's proof or census, and no additional computation was run to check the other PR.

The completion refresh checked this account's nine open PRs created since the CLAIM at 2026-10-23T13:55:07Z: #5245 through #5253, including this PR. The others concern qubit repair, finite geometry, downset inertia, symplectic spreads, diagonal completions, a projective design and odd-type fitting ranks. They do not study this determinant-image quantity. No shared-number disagreement was found. Sparse historical notes and unpublished work remain outside complete overlap coverage.

CI notes

Both hosted checks passed at exact contribution head 43142bedc7cb5c0f1f73ff3fea6d9bfd237949ee in Registered research packages run 39966509666: planning in 4m51s and registered package replay in 1m11s. Run metadata confirms completed status, success and the matching head. The replay executes both warning-strict normal and optimized registered commands. The checks were read once at completion.

All required local registered, committed-byte/source-history, export-integrity and Git-free replay checks also passed. Hosted CI certifies execution of the finite checks, not independent mathematical acceptance of the general theorem. The limitations and unrun obligations above remain unchanged.

Merge danger

Door: two-way. Blast radius: package.

Only five files in one new opt-in research package are added. Shared code, frozen predecessors and selected-frontier records are unchanged.